Preface ..................................................... IX
Acknowledgments ............................................. XI
1 Introduction ................................................. 1
1.1 Heat and Fluid Flows in Materials Science and
Engineering ............................................. 1
1.2 Overview of the Present Work ............................ 3
2 Mathematical Description of Physical Phenomena in
Thermofluid Dynamics ......................................... 7
2.1 Conservation Equations for Continuum Media .............. 7
2.1.1 Conservation of Mass ............................. 8
2.1.2 Conservation of Momentum ......................... 9
2.1.3 Energy Conservation Equation .................... 11
2.1.4 Conservation of Chemical Species ................ 14
2.1.5 Boussinesq Approximation ........................ 15
2.1.6 Unified Form of Conservation Equations .......... 16
2.1.7 Nondimensional Form of Conservation Equations ... 17
2.1.8 Short Summary ................................... 18
2.2 Boundary and Initial Conditions ........................ 19
2.2.1 Heat Transfer ................................... 20
2.2.2 Solutal Transfer ................................ 21
2.2.3 Fluid Dynamics .................................. 22
2.3 Conservation Equations in Electromagnetics ............. 24
2.3.1 Maxwell Equations ............................... 24
2.3.2 Induction and Poisson Equations ................. 25
2.3.3 An Example of a Low Magnetic Reynolds Number
Approximation: Rotating Magnetic Field .......... 28
3 Discretization Approaches and Numerical Methods ............. 39
3.1 The Finite Difference Method ........................... 40
3.1.1 Introduction .................................... 40
3.1.2 Approximation Schemes ........................... 41
3.1.3 Example of Conservative Property of FDM ......... 44
3.1.4 Discretization Schemes of Unsteady Equations .... 46
3.1.5 Example of Unsteady Diffusion Equation .......... 50
3.2 The Finite Volume Method ............................... 52
3.2.1 Basic Concept ................................... 52
3.2.2 Interpolation Schemes ........................... 54
3.2.3 Linearized Form of Discretized Conservation
Equation ........................................ 63
3.2.4 Treatment of Source Terms ....................... 66
3.2.5 Boundary Conditions ............................. 67
3.2.6 Comparative Study of Schemes for One-
Dimensional Convection/Diffusion Problem ........ 69
3.3 Solution of Linear Equation Systems .................... 73
3.3.1 Direct Methods .................................. 73
3.3.2 Iterative Methods ............................... 75
3.3.3 Residuals and Convergence ....................... 79
3.3.4 Multigrid Method ................................ 80
3.3.5 Illustration of Iterative Methods ............... 81
4 Calculations of Flows with Heat and Mass Transfer ........... 85
4.1 Solution of Incompressible Navier-Stokes Equations ..... 85
4.2 Pressure and Velocity Coupling: SIMPLE Family .......... 86
4.2.1 SIMPLE .......................................... 86
4.2.2 SIMPLER ......................................... 90
4.2.3 SIMPLE with Collocated Variables Arrangement .... 91
4.3 Illustrations of Schemes for Flow with Heat Transfer ... 94
4.4 Complex Geometry Problems on Fixed Cartesian Grids ..... 98
4.4.1 Immersed Boundary Methods ....................... 99
4.4.2 Cartesian Grid Methods ......................... 102
4.4.3 Immersed Surface Reconstruction ................ 104
4.4.4 Illustration of Continuous-Forcing IBM ......... 107
5 Convection-Diffusion Phase-Change Problems ................. 115
5.1 Some Aspects of Solidification Thermodynamics ......... 117
5.1.1 One-Component Melts ............................ 118
5.1.2 Binary Alloys .................................. 120
5.1.3 Interface and Equilibrium ...................... 125
5.2 Modeling of Macroscale Phase-Change Phenomena ......... 129
5.2.1 Heat Transfer in Phase-Change Systems: Fixed
and Moving Grids ............................... 130
5.2.2 Mathematical Models of a Binary Alloy
Solidification ................................. 137
5.2.3 Closure Relations for the Volume Fraction of
Liquid ......................................... 144
5.3 Turbulent Solidification .............................. 148
5.3.1 Review of Unsteady RAN S Modeling of
a Solidification ............................... 148
5.3.2 Conditions for the DNS of Convection-Driven
Solidification ................................. 151
5.4 Microscale Phase-Change Phenomena ..................... 152
5.4.1 Basic Modeling Concepts ........................ 154
5.4.2 Modified Cellular Automaton Model .............. 157
5.4.3 Virtual Interface Tracking Model ............... 161
5.5 Modeling of Crystal Growth ............................ 162
5.5.1 Modeling Approaches ............................ 163
5.5.2 RMF Control of Crystal Growth .................. 165
5.5.3 Model Formulation and Validation Case .......... 166
5.5.4 VGF-RMF Crystal Growth ......................... 173
5.6 Melting of Pure Calium under the Influence of
Natural Convection .................................... 180
5.6.1 State of Modeling .............................. 180
5.6.2 Model and Numerical Description ................ 182
5.6.3 Results and Discussions ........................ 183
6 Application I: Spin-Up of a Liquid Metal in Cylindrical
Cavities ................................................... 191
6.1 Spin-Up of Isothermal Flow Driven by a Rotating
Magnetic Field ........................................ 193
6.1.1 Model Formulation .............................. 193
6.1.2 Governing Equations and Characteristic
Scales ......................................... 196
6.1.3 Numerical Techniques and Code Validation ....... 197
6.1.4 The Physical Nature of Axisymmetric
Instability .................................... 199
6.1.5 Numerical Results .............................. 201
6.1.6 Discussions .................................... 210
6.1.7 Short Summary .................................. 211
6.2 Impact of Buoyancy Force on Spin-Up Dynamics .......... 212
7 Application II: Laminar and Turbulent Flows Driven by an
RMF ........................................................ 221
7.1 Laminar Flows: State of the Art ....................... 221
7.1.1 Problem Formulation ............................ 221
7.1.2 Numerical Method ............................... 224
7.1.3 Numerical Results .............................. 225
7.1.4 Discussion ..................................... 233
7.1.5 Short Summary .................................. 235
7.1.6 Estimation of Critical Taylor Number ........... 235
7.2 Turbulent Flows ....................................... 236
7.2.1 Axisymmetric Numerical Simulations ............. 237
7.2.2 RANS: k - ω Turbulence Model ................... 242
8 Application III: Contactless Mixing of Liquid Metals ....... 249
8.1 Mixing under Zero-Gravity Conditions .................. 249
8.1.1 Problem Formulation and Main Simplifications ... 250
8.1.2 Numerical Scheme and Validation Tests .......... 256
8.1.3 Numerical Results .............................. 258
8.1.4 Discussion of Different Mixing Scenarios ....... 269
8.1.5 Short Summary .................................. 272
8.2 The Impact of Gravity on Mixing ....................... 273
9 Application IV: Electromagnetic Control of Binary Metal
Alloys Solidification ...................................... 277
9.1 Control of a Binary Metal Alloy Solidification by
Use of Alternating Current Fields ..................... 277
9.1.1 Control of Unidirectional Solidification of
Al-Si Alloy by Use of RMF ...................... 278
9.1.2 Control of Side Cooled Systems by Use of RMF
and TMF ........................................ 285
9.2 Control of Solidification by Use of Steady
Electromagnetic Fields ................................ 294
9.2.1 Problem and Model Formulation .................. 295
9.2.2 Validation Test Cases .......................... 301
9.2.3 Numerical Results .............................. 303
9.3 The Impact of a Steady Electrical Current on
Unidirectional Solidification ......................... 310
9.3.1 Problem and Model Formulation .................. 311
9.3.2 Numerical Results and Discussions .............. 314
9.4 The Impact of an Electric Field on the Shape of
a Dendrite ............................................ 317
9.4.1 Problem and Model Formulation .................. 318
9.4.2 Scaling for Electrovortex Flows ................ 321
9.4.3 Numerical Method and Code Validation ........... 322
9.4.4 Numerical Results .............................. 324
9.5 The Impact of Parallel Applied Electric and Magnetic
Fields on Dendritic Growth ............................ 328
9.5.1 Problem and Model Formulation .................. 330
9.5.2 Numerical Results .............................. 331
References ................................................. 335
Index ...................................................... 351
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